A nice series for b^^h , base sqrt(2), by diagonalization

[bo198214]

Well, I felt it is needed to make it explicitely, that it is special because of the (even blueish enhanced) hyperlink:

The matrix power approach makes use of the established method to obtain non-integer powers (and other analytic functions) of finite matrices via diagonalization.
This is applied to the truncations of the Carleman/Bell matrix.

where the keyword "finite matrices" occur. We had that several times and since I had assumed, that I had made it clear that I'm always working on infinite matrices, that remark is surprising.

For me matrix power method means exactly what I wrote. Take the truncations \( B_N \) of an infinite matrix \( B \), apply the matrix power \( (B_N)^h \) and take the limit
\( B^h := \lim_{N\to\infty} (B_N)^h \).

Now one can apply this method at different development points, i.e. the original function \( f(x)=b^x \) is conjugated to the development point \( p \):

\( \tilde{f}(x)=f(p+x)-p \)
or written as composition:
\( \tilde{f}(x)=\tau_p^{-1}\circ f\circ \tau_p \)
then
\( f(x)=\tau_p \tilde{f} \tau_p^{-1} \)
we define the application of the matrix power method at point p by
\( f^h := \tau_p \tilde{f}^h \tau_p^{-1} \).

This is the general method.
If I now apply the matrix power method to a fixed point p, then the Bell/Carlemann matrix of \( \tilde{f} \) is lower/upper triangular. For those matrices the power of the truncation is the same as the truncation of the power.

That means the limit to infinity is just an expansion of the matrix, once computed values do not change in that process.

So we see that regular iteration is the particular case of applying the matrix power method to a fixed point.


And I really honored this method (applied to a *non-fixed point* like 0) because it can do where regular iteration fails: to be able to compute real iterates for \( b>e^{1/e} \). This also puzzles me that despite you insinst on regular iteration for \( b>e^{1/e} \).

Well, next question. "Why work with matrices if things are otherwise well known..." - I still don't claim something special. It's just my path into the matter: I came from a project, in which I compiled relations between pascal- Stirling, Euler- and other matrices operating on formal powerseries and stumbled on the possibility of iterating functions (other than that of addition, which I had already studied to some nice encounters with the zeta/eta-function)

But if you focus too much on only matrices and nothing else, such interesting relations like the convergence radius of the iterates is just out of scope for you. Because it is derived by the interelation of the limit formulas for regular iteration (which is power series free) and power series formulas for regular iteration.

I've just reread Andrew's "exact entries for slog-operator" today and found a similar matrix-discussion there: it helps to understand since I've no training in functional analysis, the bit I had was in 1972 to 77 and only in relation for the computer-courses, which were my main subject.

Wah I was born in that time

that's certainly what I should do. Hope I'll get things working/walking. And why the heck should the girls *pass*?

Well you are right, they build up a big cluster around your place.